A353989 -id:A353989 - cp's OEIS frontend

A375829 a(1) = 1; for n > 1, a(n) is the smallest unused positive number such that (a(n-1) AND a(n)) = 0 if a(n-1) is prime, otherwise (a(n-1) AND a(n)) = a(n-1), where AND is the binary AND operation.

1, 3, 4, 5, 2, 8, 9, 11, 16, 17, 6, 7, 24, 25, 27, 31, 32, 33, 35, 39, 47, 64, 65, 67, 12, 13, 18, 19, 36, 37, 10, 14, 15, 63, 127, 128, 129, 131, 20, 21, 23, 40, 41, 22, 30, 62, 126, 254, 255, 511, 1023, 2047, 4095, 8191, 8192, 8193, 8195, 8199, 8207, 8223, 8255, 8319, 8447, 256, 257, 26, 58, 59, 68, 69, 71, 48, 49, 51, 55, 119, 247, 503, 512, 513, 515, 519

Offset: 1

Scott R. Shannon, Aug 30 2024

nonn

The long term behavior of the terms is dominated by the appearance of the Mersenne primes. This is due to composite numbers appearing which are 1 less than a power of 2, i.e., their binary representation consists of all 1's. Therefore the next term must be 1 less than the next power of 2 to satisfy (a(n-1) AND a(n)) = a(n-1). This pattern repeats until such a number is prime, i.e., a Mersenne prime, at which point the next term will be 1 more than this prime, after which the terms can eventually return to smaller values.

Due to the above behavior it is unknown if all numbers eventually appear, but assuming the number of Mersenne primes is infinite, it is likely all eventually will. In the first 100 terms the fixed points are 1, 71, 463, although more likely exist.

			a(7) = 9 as a(6) = 8 = 1000_2 is not prime, and ((9 = 1001_2) AND 1000_2) = 1000_2 = 8.
a(9) = 16 as a(8) = 11 = 1011_2 is prime, and ((16 = 10000_2) AND 1011_2) = 0.

Scott R. Shannon, Table of n, a(n) for n = 1..1100

Cf. A375828, A000668, A109812, A007088, A115510, A000040, A353989.

A351691 Lexicographically earliest infinite sequence of distinct positive numbers such that, for n>2, a(n) has a common factor with a(n-1), shares a 1-bit in its binary expansion with a(n-1), has no common factor with a(n-2), and does not share a 1-bit in its binary expansion with a(n-2).

Original entry on oeis.org

1, 2, 6, 21, 161, 736, 66, 15, 145, 464, 68, 527, 155, 80, 96, 33, 143, 26, 48, 165, 65, 338, 14, 133, 209, 88, 10, 35, 273, 24, 40, 295, 531, 144, 136, 1037, 305, 50, 74, 333, 129, 688, 20, 325, 299, 138, 132, 341, 1147, 1184, 384, 261, 551, 608, 72, 141, 517, 770, 18, 57, 589, 1798, 34, 8313

Offset: 1

Scott R. Shannon, May 26 2022

nonn

The sequence is similar to A336957 but with the addition restrictions that each new term a(n) must share a 1-bit in its binary expansion with a(n-1), while sharing no 1-bits with the binary expansion of a(n-2). To ensure the sequence is infinite each a(n) must not only have a prime factor not in a(n-1), implying no prime or prime powers can occur (see A336957), it must also have a 1-bit in its binary expansion that is a 0-bit in the binary expansion of a(n-1).

			a(5) = 161 = 10100001_2 as a(4) = 21 = 10101_2, a(3) = 6 = 110_2, and 161 is the smallest unused number that shares a factor with 21, has a 1-bit in common with 21 in their binary expansions, does not share a factor with 6, has no 1-bit in common with 6 in their binary expansions, has a prime factor not in 21, and has a 1-bit in its binary expansion that is a 0-bit in the binary expansion of 21.

Cf. A336957, A353989, A354087, A064413, A352763.

A354420 Lexicographically earliest infinite sequence of distinct positive numbers such that, for n>3, a(n) has a common factor with a(n-2), shares a 1-bit in its binary expansion with a(n-2), has no common factor with a(n-1), and does not share a 1-bit in its binary expansion with a(n-1).

Original entry on oeis.org

1, 2, 5, 18, 65, 6, 25, 4, 35, 12, 49, 8, 7, 24, 133, 10, 21, 34, 9, 22, 105, 16, 3, 20, 33, 14, 81, 38, 129, 26, 69, 40, 23, 32, 207, 304, 15, 112, 135, 56, 195, 28, 99, 136, 39, 88, 261, 50, 141, 80, 47, 64, 423, 584, 51, 76, 17, 36, 323, 44, 19, 68, 57, 70, 153, 98, 285, 194, 45, 82, 165

Offset: 1

Scott R. Shannon, May 26 2022

nonn

The sequence is similar to A098550 but with the addition restrictions that each new term a(n) must share a 1-bit in its binary expansion with a(n-2), while sharing no 1-bits with the binary expansion of a(n-1). Unlike A351691 no additional restrictions on the factors or 1-bits of a(n) are required for the sequence to be infinite. The sequence is conjectured to be a permutation of the positive integers.

			a(5) = 65 = 1000001_2 as a(4) = 18 = 10010_2, a(3) = 5 = 101_2, and 65 is the smallest unused number that shares a factor with 5, has a 1-bit in common with 5 in their binary expansions, does not share a factor with 18, has no 1-bit in common with 18 in their binary expansions.

Cf. A098550, A351691, A064413, A353990, A336957, A353989, A354087, A352763.

cp's OEIS Frontend

A375829 a(1) = 1; for n > 1, a(n) is the smallest unused positive number such that (a(n-1) AND a(n)) = 0 if a(n-1) is prime, otherwise (a(n-1) AND a(n)) = a(n-1), where AND is the binary AND operation.

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A351691 Lexicographically earliest infinite sequence of distinct positive numbers such that, for n>2, a(n) has a common factor with a(n-1), shares a 1-bit in its binary expansion with a(n-1), has no common factor with a(n-2), and does not share a 1-bit in its binary expansion with a(n-2).

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A354420 Lexicographically earliest infinite sequence of distinct positive numbers such that, for n>3, a(n) has a common factor with a(n-2), shares a 1-bit in its binary expansion with a(n-2), has no common factor with a(n-1), and does not share a 1-bit in its binary expansion with a(n-1).

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